First Order Second Order Laplace Transforms Additional Topics Applications, Practice
Separation of Variables
Linear
Integrating Factors (Linear)
Substitution
Exact Equations
Integrating Factors (Exact)
Linear
Constant Coefficients
Substitution
Reduction of Order
Undetermined Coefficients
Variation of Parameters
Polynomial Coefficients
Cauchy-Euler Equations
Chebyshev Equations
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Classify Differential Equations
Fourier Series
Slope Fields
Wronskian
Existence and Uniqueness
Boundary Value Problems
Euler's Method
Inhomogeneous ODE's
Resonance
Partial Differential Equations
Linear Systems
Exponential Growth/Decay
Population Dynamics
Projectile Motion
Chemical Concentration
Fluids (Mixing)
Practice Problems
Practice Exam List
Exam A1
Exam A3
Exam B2

You CAN Ace Differential Equations

17calculus > differential equations > exam A1

### Differential Equations Alpha List

 Boundary Value Problems Cauchy-Euler Equations Chebyshev Equations Chemical Concentration Classify Differential Equations Constant Coefficients Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay First Order, Linear Fluids (Mixing) Fourier Series Inhomogeneous ODE's Integrating Factors (Exact) Integrating Factors (Linear) Laplace Transforms Linear Systems Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance Second Order, Linear Separation of Variables Shifting Theorems Slope Fields Solve Initial Value Problems Square Wave Substitution Undetermined Coefficients Unit Impulse Function Unit Step Function Variation of Parameters Wronskian

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Differential Equations - Exam A1

This page contains a complete differential equations exam with worked out solutions to all problems.

 Exam Details Tools Downloads Time 1 hour Calculators no Questions 12 Formula Sheet(s) none Total Points 100 Other Tools none download one page list of these questions

Instructions:
- This exam is in five main parts, labeled sections 1-5, with different instructions for each section.
- For each problem, correct answers are worth 1 point. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions.
- Correct notation counts (i.e. points will be taken off for incorrect notation).

Section 1

Classify each of the following differential equations by order and type. Each question in this section is worth 3 points.

Question 1

$$\displaystyle{ y \frac{d^2y}{dx^2} + 4\frac{dy}{dx} = \sin(x) }$$

solution

Question 2

$$\displaystyle{ \frac{d^3 y}{dy^{3}}+4\sin(x)~y = 0 }$$

solution

Section 2

Solve the following initial value problems. Each question in this section is worth 5 points.

Question 3

$$y'+2y=0; ~~~ y(0)=1$$

solution

Question 4

$$y'+ty=t;~~~y(0)=2$$

solution

Section 3

Find the general solution to each of the following. Each question in this section is worth 7 points.

Question 5

$$y'=(x+1)y$$

solution

Question 6

$$y'=y-y^2$$

solution

Section 4

Determine if the following equations are exact. If not, calculate the integrating factor to make them exact and then solve each equation. Each problem in this section is worth 10 points.

Question 7

$$2xy^2 + 4x^3 + (2x^2y +4y^3)y' = 0$$

solution

Question 8

$$y+2e^x + (1+e^{-x})y' = 0$$

solution

Section 5

Solve the following problems.

Question 9

[5 points] Show that $$y(x) = \tanh(x+a)$$ satisfies $$y'=1-y^2$$.

solution

Question 10

[10 points] Determine the stability of each critical point for the differential equation $$y'=y^2(y^2-1)$$.

solution

Question 11

[20 points] A man jumps from an airplane from a height of 1000 feet. If the force due to air resistance is proportional to the square of the velocity (with proportionality constant $$\mu$$), determine his velocity at any time. ( $$g = 32~ft/sec^2; \mu = 0.0008~ft^{-1}$$ )

solution

Question 12

[ 15 points ] A 100 gallon tank is being filled with a solution containing 100 grams per gallon of chlorine. The tank is being filled at the rate of 10 gallons per minute. Initially the tank contains 20 gallons with chlorine content of 200 grams. The well-mixed solution leaves the tank at a rate of 5 gallons per minute. Set up but do not solve the initial value problem that models this. If they fail to stop the flow in when the tank is full, what will the eventual chlorine content be?

solution